'Organ' meant 'tool' in Greek — from PIE *werg- (to work). Each body part is an instrument doing its job.
A part of the body that has a specific vital function, such as the heart, liver, or lungs; also a large musical instrument with pipes supplied with air from bellows.
From Old French 'organe,' from Latin 'organum' (instrument, tool, organ of the body, musical instrument), from Greek 'organon' (tool, instrument, bodily organ, musical instrument), from PIE *werǵ- (to work, to do). The Greek word meant any instrument or tool used to accomplish work. Its application to body parts reflects the ancient view that each organ is an instrument performing a specific task. The musical instrument sense
The words 'organ' and 'work' are distant cousins. Greek 'organon' (instrument) comes from PIE *werǵ- (to work), the same root that gave English 'work' via Germanic and Greek 'ergon' (work) — whence 'energy,' 'ergonomic,' and 'surgery' (literally hand-work). An organ is, at root, a thing that works.